Editing Integral curve

{{Short description|Term in mathematics}}
{{distinguish|Curve integral}}

In [[mathematics]], an '''integral curve''' is a [[parametric curve]] that represents a specific solution to an [[ordinary differential equation]] or system of equations.

==Name==
Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In [[physics]], integral curves for an [[electric field]] or [[magnetic field]] are known as ''[[field line]]s'', and integral curves for the [[velocity field]] of a [[fluid]] are known as [[Streamlines, streaklines, and pathlines|''streamlines'']]. In [[dynamical systems theory|dynamical systems]], the integral curves for a differential equation that governs a [[dynamical system|system]] are referred to as [[trajectory|''trajectories'']] or [[orbit (dynamics)|''orbits'']].

==Definition==
Suppose that {{math|'''F'''}} is a static [[vector field]], that is, a [[vector-valued function]] with [[Cartesian coordinate system|Cartesian coordinates]] {{math|(''F''<sub>1</sub>,''F''<sub>2</sub>,...,''F''<sub>''n''</sub>)}}, and that {{math|'''x'''(''t'')}} is a [[parametric curve]] with Cartesian coordinates {{math|(''x''<sub>1</sub>(''t''),''x''<sub>2</sub>(''t''),...,''x''<sub>''n''</sub>(''t''))}}. Then {{math|'''x'''(''t'')}} is an '''integral curve''' of {{math|'''F'''}} if it is a solution of the [[autonomous system (mathematics)|autonomous system]] of ordinary differential equations,

<math display="block">\begin{align}
\frac{dx_1}{dt} &= F_1(x_1,\ldots,x_n) \\ 
&\;\, \vdots \\
\frac{dx_n}{dt} &= F_n(x_1,\ldots,x_n).
\end{align}
</math>

Such a system may be written as a single vector equation,

<math display="block">\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).</math>

This equation says that the vector tangent to the curve at any point {{math|'''x'''(''t'')}} along the curve is precisely the vector {{math|'''F'''('''x'''(''t''))}}, and so the curve {{math|'''x'''(''t'')}} is tangent at each point to the vector field '''F'''.

If a given vector field is [[Lipschitz continuous]], then the [[Picard–Lindelöf theorem]] implies that there exists a unique flow for small time.

==Examples==
[[Image:Slope Field.png|thumb|250px|Three integral curves for the [[slope field]] corresponding to the differential equation {{math|1=''dy'' / ''dx'' = ''x''<sup>2</sup> − ''x'' − 2}}.]]
If the differential equation is represented as a [[vector field]] or [[slope field]], then the corresponding integral curves are [[tangent]] to the field at each point.

==Generalization to differentiable manifolds==
===Definition===

Let {{math|''M''}} be a [[Banach manifold]] of class {{math|''C''<sup>''r''</sup>}} with {{math|''r'' ≥ 2}}. As usual, {{math|T''M''}} denotes the [[tangent bundle]] of {{math|''M''}} with its natural [[projection (mathematics)|projection]] {{math|''π''<sub>''M''</sub> : T''M'' → ''M''}} given by

<math display="block">\pi_M : (x, v) \mapsto x.</math>

A vector field on {{math|''M''}} is a [[Fiber bundle#Sections|cross-section]] of the tangent bundle {{math|T''M''}}, i.e. an assignment to every point of the manifold {{math|''M''}} of a tangent vector to {{math|''M''}} at that point. Let {{math|''X''}} be a vector field on {{math|''M''}} of class {{math|''C''<sup>''r''−1</sup>}} and let {{math|''p'' ∈ ''M''}}. An '''integral curve''' for {{math|''X''}} passing through {{math|''p''}} at time {{math|''t''<sub>0</sub>}} is a curve {{math|''α'' : ''J'' → ''M''}} of class {{math|''C''<sup>''r''−1</sup>}}, defined on an [[interval (mathematics)|open interval]] {{math|''J''}} of the [[real line]] {{math|'''R'''}} containing {{math|''t''<sub>0</sub>}}, such that

<math display="block">\begin{align}
\alpha(t_0) &= p; \\
\alpha' (t) &= X (\alpha (t)) \text{ for all } t \in J.
\end{align}</math>

===Relationship to ordinary differential equations===

The above definition of an integral curve {{math|''α''}} for a vector field {{math|''X''}}, passing through {{math|''p''}} at time {{math|''t''<sub>0</sub>}}, is the same as saying that {{math|''α''}} is a local solution to the ordinary differential equation/initial value problem

<math display="block">\begin{align}
\alpha(t_0) &= p; \\
\alpha' (t) &= X (\alpha (t)).
\end{align}</math>

It is local in the sense that it is defined only for times in {{math|''J''}}, and not necessarily for all {{math|''t'' ≥ ''t''<sub>0</sub>}} (let alone {{math|''t'' ≤ ''t''<sub>0</sub>}}). Thus, the problem of proving the existence and uniqueness of integral curves is the same as that of finding solutions to ordinary differential equations/initial value problems and showing that they are unique.

===Remarks on the time derivative===

In the above, {{math|''α''&prime;(''t'')}} denotes the derivative of {{math|''α''}} at time {{math|''t''}}, the "direction {{math|''α''}} is pointing" at time {{math|''t''}}. From a more abstract viewpoint, this is the [[Fréchet derivative]]:

<math display="block">(\mathrm{d}_t\alpha) (+1) \in \mathrm{T}_{\alpha (t)} M.</math>

In the special case that {{math|''M''}} is some [[open subset]] of {{math|'''R'''<sup>''n''</sup>}}, this is the familiar derivative

<math display="block">\left( \frac{\mathrm{d} \alpha_1}{\mathrm{d} t}, \dots, \frac{\mathrm{d} \alpha_n}{\mathrm{d} t} \right),</math>

where {{math|''α''<sub>1</sub>, ..., ''α''<sub>''n''</sub>}} are the coordinates for {{math|''α''}} with respect to the usual coordinate directions.

The same thing may be phrased even more abstractly in terms of [[induced homomorphism|induced maps]]. Note that the tangent bundle {{math|T''J''}} of {{math|''J''}} is the [[Fiber bundle#Trivial bundle|trivial bundle]] {{math|''J'' × '''R'''}} and there is a [[canonical form|canonical]] cross-section {{math|''ι''}} of this bundle such that {{math|1=''ι''(''t'') = 1}} (or, more precisely, {{math|(''t'', 1) ∈ ''ι''}}) for all {{math|''t'' ∈ ''J''}}. The curve {{math|''α''}} induces a [[bundle map]] {{math|''α''<sub>∗</sub> : T''J'' → T''M''}} so that the following diagram commutes:

:[[Image:CommDiag TJtoTM.png]]

Then the time derivative {{math|''α''&prime;}} is the [[function composition|composition]] {{math|1=''α''&prime; = ''α''<sub>∗</sub> <small>o</small> ''ι'', and ''α''&prime;(''t'')}} is its value at some point&nbsp;{{math|''t'' ∈ ''J''}}.

== References ==
{{refbegin}}
* {{cite book | authorlink=Serge Lang | last=Lang | first=Serge | title=Differential manifolds | publisher=Addison-Wesley Publishing Co., Inc. | location=Reading, Mass.&ndash;London&ndash;Don Mills, Ont. | year=1972 }}
{{refend}}
{{Manifolds}}

[[Category:Differential geometry]]
[[Category:Ordinary differential equations]]